Cut the knot: learn to enjoy mathematics
A math books store at a unique math study site. Learn to enjoy mathematics.
Google
Web CTK
Best sites for teachers
Sites for teachers
Sites for parents
Terms of use
Awards

Interactive Activities
CTK Exchange
CTK Insights - a blog

Games & Puzzles
What Is What
Arithmetic/Algebra
Geometry
Probability
Outline Mathematics
Make an Identity
Book Reviews
Eye Opener
Analog Gadgets
Inventor's Paradox
Did you know?...
Proofs
Math as Language
Things Impossible
Visual Illusions
My Logo
Math Poll
Cut The Knot!
MSET99 Talk
Other Math sites
Front Page
Movie shortcuts
Personal info
Reciprocal links
Privacy Policy

Guest book
News sites

Recommend this site

Best sites for teachers
Sites for teachers
Sites for parents

Education & Parenting

Manifesto: what CTK is about Search CTK Buying a book is a commitment to learning Table of content Things you can find on CTK Chronology of updates Email to Cut The Knot Recommend this page

7 = 2 + 5 Sangaku

A slew of sangaku problems deal with chains of inscribed circles, see, for example, Steiner's sangaku. The elegant sangaku below is the simplest in the collection by Fukagawa and Pedoe (1.8.6), yet it gives a chance to discuss a formula that was not used so far anywhere else at the site.

 

 

The points T, A, B are collinear and TA = 2r and TB = 2s. The circles C1(r) and C2(s) have diameters TA and TB respectively. The circle O1(r1) touches AB, touches C1(r) externally and C2(s) internally, and we then form a chain of contact circles Oi(ri) (i = 1, 2, ...) as shown, all touching C1(r) externally and C2(s) internally. Show that

  7 / r4 = 2 / r7 + 5 / r1.

Solution

Copyright © 1996-2008 Alexander Bogomolny

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This sangaku dates from 1814 and was written in the Gumma prefecture.

 

The derivation of a more traditional formula for the common chain of circles inscribed in an arbelos

  rn = ts / (n² + t + t²).

is easily adapted to the present case. The result is the following formula

(*) rn = 4ts / ((2n - 1)² + 4t + 4t²),

where t = r / (s - r). Denoting S = 4t + 4t² we can write

 
  • r1 = 4s / (1 + S),
  • r4 = 4s / (7² + S),
  • r7 = 4s / (13² + S),

which we plug into the required identity

  7 / r4 = 2 / r7 + 5 / r1

getting

  7(7² + S) = 2(13² + S) + 5(1 + S)

which simplifies to

  7·7² = 2·13² + 5.

As one can easily verify, both sides are equal to 343.

Sangaku

  1. Sangaku: Reflections on the Phenomenon
  2. Critique of My View and a Response
  3. 7
  4. 3-4-5 Triangle by a Kid
  5. 7
  6. A 49th Degree Challenge
  7. A Geometric Mean Sangaku
  8. A Hard but Important Sangaku
  9. A Sangaku: Two Unrelated Circles
  10. A Sangaku by a Teen
  11. A Sangaku Follow-Up on an Archimedes' Lemma
  12. A Sangaku with an Egyptian Attachment
  13. A Sangaku with Many Circles and Some
  14. An Old Japanese Theorem
  15. Archimedes Twins in the Edo Period
  16. Arithmetic Mean Sangaku
  17. Bottema Shatters Japan's Seclusion
  18. Circles and Semicircles in Rectangle
  19. Circles in a Circular Segment
  20. Circles Lined on the Legs of a Right Triangle
  21. Equal Incircles Theorem
  22. Equilateral Triangle, Straight Line and Tangent Circles
  23. Equilateral Triangles and Incircles in a Square
  24. Five Incircles in a Square
  25. Four Hinged Squares
  26. Four Incircles in Equilateral Triangle
  27. Gion Shrine Problem
  28. Harmonic Mean Sangaku
  29. Heron's Problem
  30. In the Wasan Spirit
  31. Incenters in Cyclic Quadrilateral
  32. Japanese Art and Mathematics
  33. Malfatti's Problem
  34. Maximal Properties of the Pythagorean Relation
  35. Neuberg Sangaku
  36. Out of Pentagon Sangaku
  37. Peacock Tail Sangaku
  38. Pentagon Proportions Sangaku
  39. Pythagoras and Vecten Break Japan's Isolation
  40. Radius of a Circle by Paper Folding
  41. Review of Sacred Mathematics
  42. Sangaku ŕ la V. Thebault
  43. Sangaku and The Egyptian Triangle
  44. Sangaku in a Square
  45. Sangaku Iterations, Is it Wasan?
  46. Sangaku with 8 Circles
  47. Sangaku with Three Mixtilinear Circles
  48. Sangaku with Versines
  49. Sangakus with a Mixtilinear Circle
  50. Sequences of Touching Circles
  51. Square and Circle in a Gothic Cupola
  52. Tangent Circles and an Isosceles Triangle
  53. The Squinting Eyes Theorem
  54. Steiner's Sangaku
  55. Three Incircles In a Right Triangle
  56. Three Squares and Two Ellipses
  57. Three Tangent Circles Sangaku
  58. Triangles, Squares and Areas from Temple Geometry
  59. Two Arbelos, Two Chains
  60. Two Circles in an Angle

Copyright © 1996-2008 Alexander Bogomolny

28676879Page copy protected against web site content infringement by Copyscape


Search:
Keywords:


Latest on CTK Exchange
Math
Posted by Laura
2 messages
06:56 AM, Apr-15-08

Divisibility rules - Jargon buste ...
Posted by Carolyn
2 messages
08:35 AM, Apr-04-08

product of fractions
Posted by ke_45
3 messages
08:37 AM, May-06-08

Distance to the horizon
Posted by Monty
3 messages
04:38 PM, May-08-08

Mistake on the page (an aside, Be ...
Posted by Max
4 messages
10:28 AM, Feb-28-08

Nim Games - a query
Posted by Akash Kumar
1 messages
08:53 AM, Apr-15-08

A typo in
Posted by alexwajn
1 messages
11:36 PM, Apr-19-08